Applying the recursive rule $a_n = a_{n-1} + 3$, write the first seven terms of the sequence when $a_0 = 10$.
Answer (1)
The sequence starts with a 1 = 10 .
Apply the recursive rule a n = a n − 1 + 3 to find subsequent terms.
Calculate each term: a 2 = 13 , a 3 = 16 , a 4 = 19 , a 5 = 22 , a 6 = 25 , a 7 = 28 .
The first seven terms are: 10 , 13 , 16 , 19 , 22 , 25 , 28 .
Explanation
Understanding the Problem We are given a recursive sequence defined by the rule a n = a n − 1 + 3 , with the first term a 1 = 10 . Our goal is to find the first seven terms of this sequence. This means we need to find a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , and a 7 .
Finding the Second Term To find the second term, a 2 , we use the recursive rule with n = 2 : a 2 = a 2 − 1 + 3 = a 1 + 3 = 10 + 3 = 13 So, the second term is 13.
Finding the Third Term To find the third term, a 3 , we use the recursive rule with n = 3 : a 3 = a 3 − 1 + 3 = a 2 + 3 = 13 + 3 = 16 So, the third term is 16.
Finding the Fourth Term To find the fourth term, a 4 , we use the recursive rule with n = 4 : a 4 = a 4 − 1 + 3 = a 3 + 3 = 16 + 3 = 19 So, the fourth term is 19.
Finding the Fifth Term To find the fifth term, a 5 , we use the recursive rule with n = 5 : a 5 = a 5 − 1 + 3 = a 4 + 3 = 19 + 3 = 22 So, the fifth term is 22.
Finding the Sixth Term To find the sixth term, a 6 , we use the recursive rule with n = 6 : a 6 = a 6 − 1 + 3 = a 5 + 3 = 22 + 3 = 25 So, the sixth term is 25.
Finding the Seventh Term To find the seventh term, a 7 , we use the recursive rule with n = 7 : a 7 = a 7 − 1 + 3 = a 6 + 3 = 25 + 3 = 28 So, the seventh term is 28.
Listing the Terms Therefore, the first seven terms of the sequence are 10, 13, 16, 19, 22, 25, and 28.
Examples
Recursive sequences are used in many real-world applications, such as calculating compound interest, modeling population growth, and designing computer algorithms. For example, if you deposit $100 into a bank account that earns 3% interest each year, the amount of money in your account each year can be modeled by a recursive sequence. Understanding recursive sequences helps in predicting future values based on current values and a defined pattern.
